Correlations in Integrable Quantum Many-Body Systems

Transcription

1 Correlations in Integrable Quantum Many-Body Systems Hannover, September 2017 The second workshop of the DFG-funded research unit FOR 2316 is focussed on applications of exactly solvable models to cold gases, anyons and topological matter, non-equilibrium, dynamics and transport, as well as form factors of higher rank (super-)spin chains. Programme Tuesday Wednesday Thursday Friday 10:00 11:00 Minguzzi Sakai Ragoucy 11:30 12:30 Suzuki Patu Vernier Slavnov 14:30 15:30 Smirnov Essler Schuricht 16:00 17:00 Nirov Delfino Ardonne 17:00 18:00 Posters Posters Tsvelik 19:30 Dinner Talks are in room 267, posters and coffee breaks in room 268 of the physics building, Appelstr. 2. Kartographische Bearbeitung: Institut für Kartographie und Geoinformatik, Leibniz Universität Hannover, Datengrundlage: ATKIS-Basis-DLM der LGLN-Landesvermessung + Geobasisinformation, Hannover [

2 Talks Chains of anyons: from their structure to integrability Eddy Ardonne, Stockholm University In this talk, I will give an introduction to anyonic chains, starting from their mathematical formulation, using socalled anyon models, or more precisely, modular tensor categories. The main ingredients are the F-symbols, which can be obtained using quantum groups. The physics of anyonic chains will be discussed, focussing on the rich phase diagrams, and in particular, the integrability at special points. Quantum quenches near criticality Gesualdo Delfino, SISSA We present the theory of quantum quenches in near-critical one-dimensional systems [1,2]. Aspects that are discussed include role of interaction, role of integrability, appearance of timescales, analytic determination of one-point functions and their long time behavior. References: [1] G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) [2] G. Delfino and J. Viti, On the theory of quantum quenches in near-critical systems, J. Phys. A 50 (2017) Full counting statistics in the spin-1/2 Heisenberg XXZ chain Fabian H. L. Essler, University of Oxford The spin-1/2 Heisenberg chain exhibits a quantum critical regime characterized by quasi long-range magnetic order at zero temperature. The strength of quantum fluctuations in the ground state can be quantified by determining the probability distributions of the components of the (staggered) subsystem magnetization. Some of these are shown to exhibit scaling and the corresponding universal scaling functions are determined by free fermion methods and by exploiting a relation with the boundary sine-gordon model. Tan s contact for one-dimensional Bose and Fermi gases Anna Minguzzi, Université Grenoble Alpes A universal decay power-law of the large-momentum tails of the momentum distribution, fixed by Tan s contact coefficients, constitutes a direct signature of strong correlations in a short-range interacting quantum gas. We derive the Tan s contact of one-dimensional harmonically trapped gases, both in the case of a multicomponent Fermi gas and of a Bose gas. In the case of a multicomponent mixture, we find a direct correspondence between the value of the Tan s contact and the symmetry of the state. We show that a local density approximation (LDA) on the Bethe- Ansatz equation of state for the homogeneous gas is in excellent agreement with the results for the harmonically confined gas and predicts a scaling behavior of the Tan s contact. This provides useful analytical expressions for the dependence on the number of particles, number of components and on interaction strength. Based on work with J. Decamp, J. Jünemann, G. Lang, M. Albert, M. Rizzi, and P. Vignolo Quantum loop algebras and highest l-weight representations: General linear case Khazret Nirov, Bergische Universität Wuppertal and Institute for Nuclear Research / Moscow Universal integrability objects are determined by the choice of representations of the quantum group in the auxiliary space. We consider various representations of Borel subalgebras of quantum loop algebras to define such objects. We are especially interested in the highest l-weight representations. Here we compare the q-oscillator and prefundamental representations and argue that the latter can be obtained by tensoring the former. We also discuss how relations between the corresponding highest l-weights reproduce functional relations between integrability objects.

3 Universal relations for quantum gases in one dimension Ovidiu Patu, Institute for Space Science / Bucharest For the most general case of an one-dimensional multi-component system (bosons, fermions or an arbitrary mixture) interacting through a delta function potential and subjected to an external potential we show that the large momentum distribution of these systems behaves like n σ C σ /k 4 with C σ the contact of species σ which can be computed from the thermodynamic properties of the system (derivatives of appropriate thermodynamic potentials with respect to scattering lengths). We obtain short distance expansions for the Green s function and pair distribution function and show that the coefficients of these expansions can be expressed in terms of the density, kinetic energy and contact. In addition we derive universal thermodynamic identities for homogeneous and inhomogeneous systems connecting the pressure, total energy, trapping energy and contact. Based on work with A. Klümper. Integrable models: their Bethe vectors, scalar products and form factors Eric Ragoucy, Laboratoire de Physique Théorique / Annecy We apply the nested algebraic Bethe ansatz to integrable models. We present some explicit representations for the Bethe vectors and their scalar products, in the framework of periodic generalized models, that encompass all integrable spin chain models with (twisted) periodic boundary conditions. We review what has been (or can be) done, depending on the algebra which underlies the model (Yangian, super-yangian or quantum group). Starting from these formulas we present some general methods that allow to deduce the form factors of the models. They are of two types: the twisted scalar product and the zero modes method. Spin Drude weight in the XXZ chain at finite temperatures Kazumitsu Sakai, Tokyo University of Science In this talk, I will present our recent results on spin transport properties in the spin-1/2 XXZ spin chain at finite temperatures. Based on the functional relations among the row-to-row transfer matrices (T-systems) and their certain combinations (Y-system), the spin Drude weight and its size dependence are evaluated. This talk is based on joint work with Andreas Klümper. On the interacting Majorana chain Dirk Schuricht, Utrecht University We study the effect of interactions on Kitaev s toy model for Majorana wires. We demonstrate that even though strong repulsive interaction eventually drive the system into a Mott insulating state the competition between the (trivial) band-insulator and the (trivial) Mott insulator leads to an interjacent topological insulating state for arbitrary strong interactions. We show that the exact ground states can be obtained analytically even in the presence of interactions when the chemical potential is tuned to a particular function of the other parameters. The ground states obtained are two-fold degenerate and differ in fermion parity, as is the case with the Kitaev/Majorana chain in a topological phase. We prove that the ground state is unique in each fermion parity sector and that there exists an energy gap. Furthermore, we investigate the effect of disorder in the chemical potential. We find that, like the noninteracting system, moderate disorder supports the topological phase, while at large disorder strengths the system becomes trivial. Based on work with Niklas Gergs, Fabian Hassler, Hosho Katsura, Masahiro Takahashi. Form factors of the monodromy matrix entries in the models with gl(2 1) symmetry Nikita Slavnov, Steklov Mathematical Institute We apply the nested algebraic Bethe ansatz to the models with gl(2 1) symmetry. We obtain explicit representations for the Bethe vectors scalar products. In some particular cases we find determinant formulas for the scalar products. Starting from these formulas and using the zero modes method we obtain compact determinant representations for the form factors of the monodromy matrix entries. The latter, in their turn, lead us to determinant formulas for form factors of local operators.

4 New approach to computation of correlation functions for XXX model Fedor Smirnov, Université Pierre et Marie Curie It has been shown that using the fermionic basis the correlation functions for XXZ model on a cylinder are expressed in universal form for arbitrary Matsubara data. We use this arbitrariness in order to fix the coefficients in the expansion of operators in the fermionic basis. For XXX case this allows to compute the correlation functions up to 11 sites. The dependence of coefficients on the lattice spacing exhibits remarkable regularity. The static and the dynamical form factor expansion approach to quantum correlations Junji Suzuki, Shizuoka University We discuss a quantitative analysis on the correlation functions of spin 1/2 XXZ model based on the quantum transfer matrix. The static and the dynamical aspects of the correlations will be discussed within the framework of form factor expansions. The content of the talk is based on collaborations with M. Dugave, F Göhmann, A. Klümper and K. K. Kozlowski. SU(2)-symmetric lattice spin model with Majorana fermion excitations Alexei M. Tsvelik, Brookhaven National Lab We have constructed a 2D lattice model of spins 1/2 interacting with nearest neighbor 2- and 3-spin interactions which have gapped bulk and gapless chiral excitations on the edges. The bulk excitations are 2D Majorana fermions and 1D solitons. The 2D propagating Majorana s are bound states of fractionalized 1D solitons. The model has two phases Abelian and non-abelian topological ones. Dynamics of observables in out-of-equilibrium many-body quantum systems: the Loschmidt echo Éric Vernier, SISSA While much progress has been made over the last years in understanding the relaxation mechanisms taking place in the non-equilibrium dynamics of quantum many-body systems, very few analytical results exist concerning the full time evolution of physical observables, however urgently called for by the ongoing advances in cold-atomic experiments as well as by the recently emerged subject of dynamical phase transitions. A reason for this fact is that despite the existence of prototypical integrable models, the time dynamics involves contributions of arbitrarily excited eigenstates of the Hamiltonian which render exact calculations prohibitively difficult. In this seminar I will present a method to tackle the exact-time dynamics of quantum observables in a prototypical interacting integrable quantum many-body system, the Heisenberg XXZ spin chain, starting with a simple observable that is the Loschmidt echo (or quantum fidelity). The latter measures the overlap between the system s state at a given time and its initial state, and has attracted a renewed interest recently in the context of dynamical phase transitions, which it signals through its non-analyticities as a function of the time. Using a reformulation of the problem in terms of an auxiliary boundary quantum transfer matrix, the Loschmidt echo is written as the solution of a set of Non Linear Integral Equations, which allows for its exact determination at arbitrarily large time. This method overcomes the time limitations experienced by numerical approaches, and allows to for an analytic approach to dynamical transitions. I will further discuss perspectives concerning the classification of integrable initial states allowing for such exact computations, as well as the extension of this method to the study of more general physical observables. This is based on L Piroli, B Pozsgay, E Vernier, J. Stat. Mech. (2017) , as well as some ongoing work.

5 Posters Transport properties of integrable quantum systems Christina Ballnus, Bergische Universität Wuppertal Signatures of non-abelian anyons in the thermodynamics of an interacting fermion model Daniel Borcherding, Leibniz Universität Hannover Functional equation for the density matrix of an inhomogeneous finite Heisenberg chain Simone Faldella, Leibniz Universität Hannover Spectral flow for an integrable staggered superspin chain Konstantin Hobuß, Leibniz Universität Hannover Lattice approach to integrable quantum field theories and applications Artur Hutsaliuk, Bergische Universität Wuppertal Dynamical phase transitions in one-dimensional topological insulators Philipp Jaeger, TU Kaiserslautern The Anderson impurity model with modified density of states Yahya Öz, Bergische Universität Wuppertal Long range correlations generated by phase separation. Exact results from field theory Alessio Squarcini, Max Planck Institute for Intelligent Systems A systematic formulation of efficient thermodynamics of 1D systems through Bäcklund relations Eyzo Stouten, Bergische Universität Wuppertal Quantum Integrable Systems and Generalized Hydrodynamics Andrew Urichuk, University of Manitoba Functional equations for the density matrix of finite-size face models Daniel Westerfeld, Leibniz Universität Hannover